Properties

Label 71957.a.71957.1
Conductor $71957$
Discriminant $71957$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = 3x^3 + 5x^2 + 2x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = 3x^3z^3 + 5x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^4 + 14x^3 + 21x^2 + 10x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 5, 3]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 5, 3], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 10, 21, 14, 2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(71957\) \(=\) \( 47 \cdot 1531 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(71957\) \(=\) \( 47 \cdot 1531 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(132\) \(=\)  \( 2^{2} \cdot 3 \cdot 11 \)
\( I_4 \)  \(=\) \(5001\) \(=\)  \( 3 \cdot 1667 \)
\( I_6 \)  \(=\) \(-172491\) \(=\)  \( - 3 \cdot 11 \cdot 5227 \)
\( I_{10} \)  \(=\) \(9210496\) \(=\)  \( 2^{7} \cdot 47 \cdot 1531 \)
\( J_2 \)  \(=\) \(33\) \(=\)  \( 3 \cdot 11 \)
\( J_4 \)  \(=\) \(-163\) \(=\)  \( -163 \)
\( J_6 \)  \(=\) \(4389\) \(=\)  \( 3 \cdot 7 \cdot 11 \cdot 19 \)
\( J_8 \)  \(=\) \(29567\) \(=\)  \( 29567 \)
\( J_{10} \)  \(=\) \(71957\) \(=\)  \( 47 \cdot 1531 \)
\( g_1 \)  \(=\) \(39135393/71957\)
\( g_2 \)  \(=\) \(-5857731/71957\)
\( g_3 \)  \(=\) \(4779621/71957\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-2 : 1 : 1)\) \((1 : 2 : 1)\) \((-2 : 0 : 3)\) \((-2 : -1 : 3)\) \((1 : -5 : 1)\) \((-2 : 8 : 1)\)
\((1 : 8 : 2)\) \((1 : -21 : 2)\) \((-7 : -125 : 10)\) \((-7 : 168 : 10)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-2 : 1 : 1)\) \((1 : 2 : 1)\) \((-2 : 0 : 3)\) \((-2 : -1 : 3)\) \((1 : -5 : 1)\) \((-2 : 8 : 1)\)
\((1 : 8 : 2)\) \((1 : -21 : 2)\) \((-7 : -125 : 10)\) \((-7 : 168 : 10)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-2 : -1 : 3)\) \((-2 : 1 : 3)\) \((1 : -7 : 1)\) \((1 : 7 : 1)\) \((-2 : -7 : 1)\) \((-2 : 7 : 1)\)
\((1 : -29 : 2)\) \((1 : 29 : 2)\) \((-7 : -293 : 10)\) \((-7 : 293 : 10)\)

magma: [C![-7,-125,10],C![-7,168,10],C![-2,-1,3],C![-2,0,3],C![-2,1,1],C![-2,8,1],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-21,2],C![1,-5,1],C![1,-1,0],C![1,0,0],C![1,2,1],C![1,8,2]]; // minimal model
 
magma: [C![-7,-293,10],C![-7,293,10],C![-2,-1,3],C![-2,1,3],C![-2,-7,1],C![-2,7,1],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-29,2],C![1,-7,1],C![1,-1,0],C![1,1,0],C![1,7,1],C![1,29,2]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.435811\) \(\infty\)
\((-1 : 1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.444812\) \(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.244833\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.435811\) \(\infty\)
\((-1 : 1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.444812\) \(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.244833\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3xz^2 - z^3\) \(0.435811\) \(\infty\)
\((-1 : 1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + 3z^3\) \(0.444812\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + z^3\) \(0.244833\) \(\infty\)

2-torsion field: 6.2.4605248.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.035808 \)
Real period: \( 16.99797 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.608667 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(47\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 47 T^{2} )\)
\(1531\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 22 T + 1531 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);